## The Grimoire Project

The Grimoire is a textbook on commutative algebra for advanced master students and graduate students that I am currently writing. Some of its last chapters could be interesting also to more advanced researchers. It is written in French. Here you can download the current version :

• Grimoire d'Algèbre Commutative

Status Update :

• November 8, 2014 : work on the Grimoire project has resumed. Lectures Bélier, Taureau et Gémeaux are complete; currently working on Cancer.

• December 7, 2014 : Lectures Bélier, Taureau, Gémeaux et Cancer are complete; currently working on Lion.

• March 6, 2015 : Lectures Bélier, Taureau, Gémeaux, Cancer and Lion are complete; currently working on Vierge.

• May 23, 2015 : The Grimoire Project enters Summer recess : until the end of September I will be working on a different project, so there will be not much action for the Grimoire : I might make the occasional correction, and maybe toy with the index once and then, but other than that don't expect much until October. The Vierge lecture is not quite finished yet, but it is well advanced.

• October 14, 2015 : work on the Grimoire project has resumed. Currently working on Vierge.

• October 23, 2015 : Lectures Bélier, Taureau, Gémeaux, Cancer, Lion et Vierge are complete; currently working on Balance.

• November 1, 2015 : I've corrected the proof of Zariski's Main Theorem, that was confusing and slightly broken in various places. Please download the revised version, and keep on sending me your remarks. I've also added a result on UFDs with some applications, at the end of section 11.4. Now, back to Balance.

• December 15, 2015 : Lectures Bélier, Taureau, Gémeaux, Cancer, Lion, Vierge and Balance are complete; currently working on Scorpion. I've also made some small corrections to the proof of Swan's theorem in Lecture 5 (Lion), and I've added some exercices at the end of section 6.5.

• February 1, 2016 : I've taken the important decision to add a section (a revised section 5.5 that I'm writing now) on the language of sheaves and a few basic notions of scheme theory. This has entailed some reorganization in lectures Lion, Vierge and Balance. The reason to include this material is that eventually I'm planning to give a presentation of a valuation-theoretic proof due (essentially) to Fujiwara of Gruson-Raynaud's important theorem on flattening of morphisms by admissible blow-ups. This proof is more elementary than Gruson and Raynaud's original argument, but of course to properly state the theorem and to give a clean presentation of its proof it is convenient to be able to manipulate some non-affine schemes (such as blow-ups).

• April 7, 2016 : I've finally completed the digression on sheaves and schemes announced in February; it is longer than originally planned and is now split over two sections : the new section 3.4 on sheaves and the new section 5.5 on schemes. This entailed some rearranging of material around the text. Now I'm ready to return to Scorpion.

• April 29, 2016 : The Grimoire Project enters Summer recess : until late September I will be working on a different project, so not much will happen on this page. Scorpion is not quite finished, but is well advanced. See you in a few months!

• November 21, 2016 : The Grimoire Project is active again! I was very busy until now, but at last I am free. I've just started a new section in Scorpion, dedicated to the real spectrum of a ring : it doesn't contain much at present, but watch it grow!

• January 16, 2017 : Lectures Bélier, Taureau, Gémeaux, Cancer, Lion, Vierge, Balance and Scorpion are complete; next up is Sagittaire.

• April 13, 2017 : Lectures Bélier, Taureau, Gémeaux, Cancer, Lion, Vierge, Balance, Scorpion and Sagittaire are complete. I've also made several minor fixes in earlier parts of the text, and added several new entries to the index. There will be a few further updates next week, then the Grimoire Project will enter Summer recess.

• December 12, 2017 : work on the Grimoire Project has finally resumed! Currently working on Capricorne.

• March 4, 2018 : today the Grimoire Project has reached an important milestone : the proofs of both the Gruson-Raynaud theorem and of elimination of quantifiers for fields with non trivial valuations are complete : see theorems 10.57 and 10.62 (I give a geometric form of elimination of quantifiers : see the remarks before paragraph 10.4.3). Still working on Capricorne : section 4 is almost done, and it remains (only) to write section 5.

• Mai 5, 2018 : Lectures Bélier, Taureau, Gémeaux, Cancer, Lion, Vierge, Balance, Scorpion, Sagittaire and Capricorne are complete. Today the Grimoire also enters officially into summer recess. I might do some little cleaning up, and also add a few entries to the index, but that will be it until October at the earliest, I expect.

• June 27, 2018 : I took advantage of some free time to make a thorough reorganization of the last 4 chapters of the Grimoire (and I've also added an exercise in what is now section 9.2). The upshot is that, for the time being, the Grimoire is complete only up to Sagittaire, so in a sense this is a step backward, but it's only in order to prepare a leap forward that I'll take after the summer : then hopefully I'll be able to complete both Capricorne and Verseau in a relatively short time. Also, you can start seeing now the general shape that Poissons will have in the end : more material is planned, but the basic scaffolding is in place.

• September 23, 2018 : work on the Grimoire Project has resumed! Currently working on the revised version of Capricorne.

• November 2, 2018 : I'm finally done with the revision of Capricorne! It took slightly longer and slightly more effort than I had anticipated, because while working on it, I realized that the previous draft was rather (and embarassingly) rough : now it is much better. Hence : lectures Bélier, Taureau, Gémeaux, Cancer, Lion, Vierge, Balance, Scorpion, Sagittaire and Capricorne are again complete, and I'll start working on Verseau in a matter of days, after I rest a little bit with a short break.

• December 30, 2018 : During the holidays I'm not working on Verseau, but I'm not quite idle either : I just finished fixing a very old and embarrassing mistake : the homological and cohomological notations for complexes were all mixed up, throughout the Grimoire. I have been aware of the problem for years, but I had never found the time and the nerve to correct this : now it's done. I'll resume working on Verseau around the second week of January : stay tuned!

• January 26, 2019 : Lectures Bélier, Taureau, Gémeaux, Cancer, Lion, Vierge, Balance, Scorpion, Sagittaire, Capricorne and Verseau are complete. I'm taking a well deserved break from the Grimoire, to take care of a bunch of long-delayed other smaller projects.

• April 22, 2019 : I've just completed a major revision of Bélier : two of the previous sections (1.3 and 1.4) have fused into a single longer one, and a new section 1.4 has been added, containing a self-contained treatment of the elementary theory of fields; moreover, some material on unique factorisation domains has been moved from section 5.2 to section 1.1.

• May 8, 2019 : today the Grimoire project enters officially into summer recess : I might add some entries to the Index, but do not expect any substantial work until October at the earliest. The last addition to the text so far is problem 12.27, detailing Nagata's example of an infinite dimensional noetherian domain. See you in a few months!

• September 29, 2019 : work on the Grimoire Project has resumed! Currently working on Poissons. The first new result is a proof of the classical Molien formula for the Hilbert-Poincaré series of the ring of polynomial invariants for the action of a finite group on a finite-dimensional vector space : see Problem 12.55.

• January 4, 2020 : Today the Grimoire reaches an important milestone : lectures Bélier, Taureau, Gémeaux, Cancer, Lion, Vierge, Balance, Scorpion, Sagittaire, Capricorne, Verseau and Poissons are complete : that's the entire Zodiac! However, this is not yet the end of the project, since I'll still make some substantial revision and improvements to some earlier chapters : stay tuned!

• February 21, 2020 : Today I've completed the first of a planned series of major revisions of the Grimoire : the objective of this revision was to include the proof of a classical result, stating that one can desingularize an integral noetherian scheme of dimension one (that is, a curve) by a canonical sequence of blowing-up morphisms centered at the singular loci, provided that the normalization map of the curve is a finite morphism (this is really the minimal condition required for the existence of such a desingularizing sequence of blowing-ups). This theorem is now contained in section 7.4 : see theorem 7.58. The proof in itself is not so long, but it uses a good part of the material developed in the first 7 lectures, including Zariski's Main Theorem; I had to add a certain amount of preliminaries concerning schemes, and that's why sections 5.4 and 5.5 now are considerably longer and look rather different, and sections 5.2 and 5.3 have fused into a single section. Moreover, some scheme-theoretic foundational material that was previously scattered in lectures 10 and 11 now has been also moved and reorganized in these two sections 5.4 and 5.5.

• April 19, 2020 : The second major revision of the Grimoire is now essentially complete : this revision has invested and completely overhauled the last three lectures. So, the material that formerly occupied Poissons now has become Verseau, and the bulk of what used to be Verseau and Capricorne has been reorganized between Capricorne and Poissons, with many notable additions, and some deletion, as the new material has superseded previous treatment in some section. Moreover, section 5 of Sagittaire is now devoted to the Artin-Rees lemma (formerly a topic in Verseau), and the material concerning Huber rings is now collected in section 4. What used to be section 5 of Sagittaire, (on continuous valuations) has become the first section of Capricorne, and this lecture now ends with a section 5 dedicated to the Gruson-Raynaud theorem. The generalities on affinoid domains and adic spaces now occupy the first section of Poissons; the second section is completely new, and presents the theory of analytically noetherian rings, superseding the previous treatment of strongly noetherian rings (which are now a special case); the advantage is that analytically noetherian rings also encompass the Huber rings with a noetherian ring of definition, which were treated by Huber separately, with methods from algebraic geometry. Instead, we derive all the basic features of (universally) analytically noetherian rings -- including strict acyclicity of the Cech complexes with coefficients in the structure presheaf -- in a unified fashion, by an essentially combinatorial analysis that generalizes the original treatment of Tate. I might add some further complements later on, but essentially we're done with this second major revision : only one major revision left before we're done!

• May 9, 2020 : I am currently halfway through the last major revision. The aim of this revision is to include some material concerning abelian categories, up to and including the Freyd-Mitchell's theorem. So far I've added the basics of abelian categories, all the way to the snake lemma. This has entailed a thorough reorganization of Bélier and Taureau. The changes in Bélier are relatively minor, but Taureau is now completely different, and is wholly devoted to category theory. Right now Gémeaux is in a messy state, and at the end of the revision it will look very different; the same goes for Cancer and Lion which will be both seriously affected by this revision. Another important change is that I've spent some efforts in improving the treatment of certain set-theoretic issues having to do with the manipulation of large categories; now our set-theoretic framework is explicitly spelled out (the Grimoire officially adopts the Bernays-Goedel axiomatics for set theoretic foundations), and a certain care is applied in distinguishing between small and large categories, especially in the treatment of limits and the constructions of categories of functors. Until now, these issues could be just swept under the rug, but the proof of the Freyd-Mitchell's theorem involves handling some very large constructions, so the whole proof gets very shaky if one neglects dealing with these size issues.

• May 29, 2020 : The final major revision of the Grimoire is not finished yet, but as of today Gémeaux has again reached a stable configuration, and it will no longer change. This chapter now contains the material on tensor products and on localization of rings (including Nakayama's lemma); the main change is that now I introduce tensor products from the point of view of bimodules : this is the only reasonable way to proceed if one wants to include the case of modules over non-commutative rings, but has also the benefit of streamlining the treatment in several places, even in the commutative case. The non-commutative case has been gradually forced on me, because of the Freyd-Mitchell theorem, which embeds every abelian category fully faithfully and exactly into a category of modules over some (usually non-commutative) ring. Anyway, whereas Gémeaux is now stable, both Cancer and Lion are still in flux, though they are, one could say, structurally sound : for Cancer, only section 1 is still incomplete : this is dedicated to advanced material on abelian categories; so far it contains a complete proof of the Gabriel-Popescu theorem (my proof is somewhat simpler than the one that is found in Kashiwara-Schapira's book on "Categories and Sheaves", but I don't know if it is new : for all I know, it might be the original proof of Gabriel and/or Popescu) and also of Grothendieck's classical theorem on the existence of injective envelopes in Grothendieck categories (aka cocomplete well-powered AB5 categories); my proof here is essentially due to Mitchell, though I reconstructed it from a very incomplete and sloppy account of it found in a little book by Freyd : what I like about it, is that it doesn't require transfinite induction : Zorn's lemma suffices (well, it always does... but the switch from one way of writing things to another is not always easy or natural... and this proof is not just some transcription of Grothendieck's proof : it's really a different proof). What is missing from this section is the Freyd-Mitchell theorem, and then we will be done! Likewise, in Lion, only section 1 will change : it's about homotopies and resolutions, but so far it's only for chain complexes of modules, and I will do it for chain complexes in any abelian category; the same goes for sections 7.5 and 8.5. But for now I'm taking a much deserved break.

• July 22, 2020 : As of today, Cancer is again stable, and will no longer change. So, section 1 of this chapter is now complete, and ends with the already announced proof of the Freyd-Mitchell theorem. The next chore is to revise section 1 of Lion, dedicated to resolutions and homotopies of complexes : currently it deals only with chain complexes of modules over a given ring, and it will be upgraded to general abelian categories. Similarly, I will need also to upgrade sections 7.5 and 8.5.

• July 28, 2020 : As of today, also Lion is again stable, and will no longer change. Only section 1 needed revising, and I also updated the introduction to this chapter. Next up would be sections 7.5 and 8.5, but I'll probably work on them a little later.

• August 10, 2020 : Another important milestone for the Grimoire : as of today, the Index is complete : it is 15 page long (14 pages for the online version that you can download) and it lists 769 items!

• August 13, 2020 : Today I've completed the revision of section 7.5. This is the section dedicated to the construction of derived functors of additive functors; previously it was only for functors between categories of modules, and now it deals with additive functors between general abelian categories : olé!

• August 21, 2020 : Today I've completed the revision of section 8.5, dedicated to double complexes. Previously it only dealt with double complexes of modules, and now it covers double complexes in any abelian category. This achieves the last major revision of the Grimoire. There remain a few small additions and minor improvements to finish up, but the end is nigh!

• October 12, 2020 : Today I've completed a thorough overhaul of the treatment of faithfully flat descent (see pages 303--310). The new version is a combination of conceptual arguments and explicit calculations with tensors; on the "conceptual" side, the main novelty is the introduction of cartesian functors (in the special situation that is needed for faithfully flat descent, so I do not develop a fully-fledged theory of fibrations). The idea of cartesian functors improves the foundations of descent theory, but is also applied later, for the proof that the category of quasi-coherent modules on Spec(A) is equivalent to that of A-modules (problem 5.57(i.b)). For this, I already used faithfully flat descent early on, but the previous treatment was very awkward, since the lack of adequate foundations forced me to proceed by checking painstakingly the commutativity of several annoying diagrams of modules... This was the part of the Grimoire which I considered the least satisfactory; now the proof looks completely different, and is very much improved.

• November 29, 2020 : Today I've finally completed the last important addition to the Grimoire : it's a generalization of the work of Buzzard and Verberkmoes which establishes that so-called stably uniform Tate-Huber rings are "sheafy" : the version of the Grimoire considers more generally arbitrary Huber rings (a Tate-Huber ring is a Huber ring that has a topologically nilpotent unit). One interesting aspect is that the proof of sheafiness in this case is achieved by the same method that the Grimoire uses for proving "sheafiness" (aka Tate acyclicity) in all the other cases : so now we have a single method for showing Tate acyclicity that works uniformly in all the currently known cases, and is essentially elementary(!) Especially, this includes the case of perfectoid rings! That's because perfectoids are stably uniform : this was the original motivation of Buzzard & Verberkmoes. Moreover, the version of the Grimoire applies even to the more general perfectoids that Gabber and me introduce in our chunky manuscript on "Foundations for Almost Ring Theory" : olé!

• December 12, 2020 : As of today, the Grimoire Project is complete! Rejoice! Shout it from the rooftops! Now comes the last phase : the final reading of the whole manuscript, which I expect will last at least 6 more months : so, you're still in time to send me your observations/corrections/encouragements. The final reading will start in a week or so, after a short and well deserved break. Thank you very much for accompanying me in this adventure!

• January 8, 2021 : The final reading of Bélier is now complete! I corrected zillions of misprints and minor (sometimes, not so minor) mistakes, and made a good number of tiny improvements : now is as perfect and sublime as it will ever get! Only 11 chapters to go!

• February 19, 2021 : The final reading of Taureau is now complete! Besides correcting countless misprints and minor mistakes, the most notable change is the addition of the construction of the quotient of an abelian category by a Serre subcategory : this is now Problem 2.101. I tried to be careful with the set-theoretic issues involved in this construction : one needs that the ambient category is well-powered; then, even with this assumption, one has to be careful in the construction of the sets of morphisms in the quotient category; this makes some arguments slightly less transparent than how one might want, but that seems to be the price to be paid in order to achieve a fully rigorous proof.

• March, 2, 2021 : The final reading of Gémeaux is now complete : halleluyah!! I corrected tons of misprints and minor mistakes, as usual, but no substantial changes were required, so this was a quickie : only two weeks for this chapter. Onward to Cancer!

• May 5, 2021 : The final reading of Cancer is now complete! This took much longer : almost exactly two months. That's partly due to all the teaching I had to do this semester, which slowed me down considerably; but it's also because this was a complex and long chapter to read and revise. Apart from the usual few tons of corrections of minor mistakes, the main changes are in section 4.3, dedicated to presheaves and sheaves. Especially, I've decided to redo some of the basic constructions for sheaves and presheaves using the language of Kan extensions : then, I had to revise also parts of Taureau, to include basic material on Kan extension (which I've added to section 2.3). Another addition is exercise 4.88(ii), proving that the presheaves and sheaves of modules (over any given ring) form a Grothendieck abelian category with a generator; in particular, this shows that such categories have enough injectives, due to a result of Grothendieck which is given in the Grimoire as theorem 4.31.

• May 14, 2021 : I've just completed the final reading of the first section of Lion; it's a long section that contains, among others, my treatment of faithfully flat descent, which has undergone a thorough revision : it was rather hard to read and some explanations were not very conceptual and rather clumsy; now it's much better. Also, I've improved the presentation of the basics of the Cech complex : previously the notation was unnecessarily cumbersome and probably confusing, so also this part is now rather better.

• June 6, 2021 : The final reading of Lion is now complete! That took almost exactly one month, which helps us to put us back in schedule, but the road ahead is still long. One notable small improvement is in the tratment of the glueing of ringed spaces : now this is presented more explicitly as a coequalizer of two glueing morphisms (lemma 5.74). Also, I've added a little exercise, showing that the presheaves and sheaves of modules over a sheaf of rings form a Grothendieck category with a generator : this completes an exercise already mentioned in the previous chapter; the proof actually reduces to that previous exercise, via the technique of tensor products of presheaves (and sheaves) of modules.